Pathfinding and mazes

CSE 3541Matt Boggus

Pathfinding – problem statement

• Finding the shortest route between two points

• How to represent the environment

• Lots of algorithms

Environment representation

• Agents need to know where they can move• Search space should represent either– Clear routes that can be traversed– Or the entire walkable surface

• Search space typically doesn’t represent:– Small obstacles or moving objects

• Most common search space representations:– Grids– Waypoint graphs– Navigation meshes

Grids

Grids

Grids as Graphs

Distance metrics

Waypoint graph

Constructing waypoint graphs

Sampling discussion and boardwork

Navigation mesh

Navigation mesh with adjacency graph

Pathfinding algorithms

• Random walk• Random trace• Breadth first search• Best first search• Dijkstra’s algorithm• A* algorithm

Evaluating graph search algorithms

• Quality of final path• Resource consumption during search– CPU and memory

• Whether it is a complete algorithm– A complete algorithm guarantees to find a path if

one exists

Random walk

• Agent takes a random step– If goal reached, then done

• Repeat procedure until goal reached

• Add intelligence– Only step in cases where distance to goal is smaller– Stuck? With a certain probability, allow a step where

distance to goal is greater

Random trace

• Agent moves towards goal– If goal reached, then done– If obstacle• Trace around the obstacle clockwise or counter-

clockwise (pick randomly) until free path towards goal

• Repeat procedure until goal reached

Random walk and trace properties

• Not complete algorithms

• Found paths are unlikely to be optimal

• Consume very little memory

Random walk and trace performance

Search algorithms

• Breadth-First• Best-First• Dijkstra’s• A* (combination of Best-First and Dijkstra)

• Nodes represent candidate paths• Keep track of numerous paths simultaneously

Graph search complete algorithms

• Logically, these algorithms use two lists: open and closed

• Open list keeps track of unexplored nodes

• Examine a node on the open list:– Check to see if it is the goal– If not, check its edges to add more nodes to the open list

• Each added node contains a reference to the node that explored it

– Place current node on closed list (mark a field as visited)

• Closed list are those that have been explored/processed and are not the goal

Implementation

1. Create start point node and push onto open list2. While open list is not empty

A. Pop node from open list (call it currentNode)B. If currentNode is the goal, break from step 2C. Create new nodes (successors nodes) for cells

around currentNode and push them onto open listD. Put currentNode onto closed list

// How to compute the solution path?

Node selection• Breadth-First processes the node that has been waiting the longest

– Uses a queue

• Best-First processes the one that is closest to the goal– Uses a priority queue –node value is distance to goal

• Dijkstra’s processes the one that is the cheapest to reach from the start cell – Uses a priority queue – node value is lowest cost from the start

• A* chooses a node that is cheap and close to the goal– Uses a priority queue – node value is a combination of Best-First and

Dijkstra’s

Breadth first search

Breadth first search in a grid

• Represent environment using a grid of integers that encode spatial information

• int grid[x,y]– Initial values:

• For starting position of search, grid[x,y] = 0• For impassable grid cells, grid[x,y] = -1• Everywhere else, grid[x,y] = INFINITY

– After the search:• grid[x,y] = distance from start

-1 if impassable INFINITY if unreachable from start

Flood-fill

• Determine the area connected to a given node in a multi-dimensional array

• Applications:– “Paintbucket” tool– Connected components– Pathfinding

Animation fromhttp://en.wikipedia.org/wiki/Flood_fill

Recursive Flood-fill codeFlood-fill (x, y, value, grid) { if(x < 0 || y < 0 || x >= XSIZE || y >= YSIZE) return; if(grid[x,y] == -1 || grid[x,y] == value) return;

grid[x,y] = value; Flood-fill(x-1,y, value, grid); Flood-fill(x+1,y, value, grid); Flood-fill(x,y+1, value, grid); Flood-fill(x,y-1, value, grid);}

Note: method is prone to stack overflow

Iterative Flood-fill algorithmFlood-fill (x, y, value, grid) { Set Q to the empty queue Add position(x,y) to Q While Q is not empty { Dequeue position p if (p.x or p.y out of bounds) continue; if (grid[p.x,p.y] == -1 || grid[p.x,p.y] == value) continue; grid[p.x,p.y] = value; Enqueue (p.x-1,p.y); Enqueue (p.x+1,p.y); Enqueue (p.x,p.y-1); Enqueue (p.x,p.y+1); }

Queue vs. Stack Flood-fills

Animations from http://en.wikipedia.org/wiki/Flood_fill

Flood-fill with varying value

Flood-fill (x, y, value, grid) { if(x < 0 || y < 0 || x >= XSIZE || y >= YSIZE)

return; if(grid[x,y] == -1 || grid[x,y] <= value)

return; grid[x,y] = value; Flood-fill(x-1,y, value+1, grid); Flood-fill(x+1,y, value+1, grid); Flood-fill(x,y+1, value+1, grid); Flood-fill(x,y-1, value+1, grid);}

Result: distance field (8 connected)

Result: distance field (8 connected)

Best first search

Suboptimal best first search

Dijkstra’s algorithm

• Disregards distance to goal– Keeps track of the cost of every path– No guessing / no search heuristic

• Computes accumulated cost paid to reach a node from the start

• Uses the cost (called the given cost) as a priority value to determine the next node that should be brought out of the open list

Dijkstra search

A* algorithm

• Admissible heuristic function that never overestimates the true cost

• To order the open list, use– Heuristic cost – estimated cost to reach the goal– Given cost – cost to reach node from the start

Final Cost = Given Cost + (Heuristic Cost * Heuristic Weight)

Heuristic weight

• Final Cost = Given Cost + (Heuristic Cost * Heuristic Weight)

• Heuristic weight controls the emphasis on the heuristic cost versus the given cost

• Controls the behavior of A*– If hw=0, final cost will be the given cost ->Dijkstra– As hw increases, it behaves more like Best-First

A* search

A* and Dijkstra comparison

More reading on pathfinding

• Amit’s A* Pages http://theory.stanford.edu/~amitp/GameProgramming/

• Beginner’s guide to pathfinding http://ai-depot.com/Tutorial/PathFinding.html

• Waypoint graphs vs. navigation meshes http://www.ai-blog.net/archives/000152.html

Mazes

Maze taken from Image-guided maze construction, Xu and Kaplan 2007, Siggraph

Simple mazes – binary grid

• Matrix of booleans• ex: 21 x 21

• Arbitrary mapping:– True = black pixel = wall– False = white pixel =

open space

Game mazes– enumerated grid

• Matrix of enumerated values : ints

• Arbitrary mapping– 0 = white pixel = open

space– 1 = black pixel = wall– 2 = health = red plus– Etc…

Procedural maze initial values

All open space – add walls All walls – add open space

Advanced algorithms can also start with artist input or an image

Maze generation – start with all walls

• Randomized depth first search– Mark current node as visited– Make list of neighbors, pick one randomly– If new node is not marked as visited, remove the

wall between it and its parent– Recurse

Maze generation – start with all walls

• Randomized Kruskal's algorithm– Randomly select a wall that joins two

unconnected open areas, and merge them– For an n x n grid there are initially n2

open areas, one per grid cell

• Randomized Prim's algorithm– Randomly pick a starting point– Keep track of walls on the boundary of the maze– Randomly select a wall that “grows” the maze

Examples

Animation examples fromhttp://en.wikipedia.org/wiki/Maze_generation_algorithm